Theorem brcogw 4831

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Proof of Theorem brcogw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1002	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A e. V )
2		simpl2 1003	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> B e. W )
3		breq2 4033	. . . . . 6 |- ( x = X -> ( A D x <-> A D X ) )
4		breq1 4032	. . . . . 6 |- ( x = X -> ( x C B <-> X C B ) )
5	3, 4	anbi12d 473	. . . . 5 |- ( x = X -> ( ( A D x /\ x C B ) <-> ( A D X /\ X C B ) ) )
6	5	spcegv 2848	. . . 4 |- ( X e. Z -> ( ( A D X /\ X C B ) -> E. x ( A D x /\ x C B ) ) )
7	6	imp 124	. . 3 |- ( ( X e. Z /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
8	7	3ad2antl3 1163	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
9		brcog 4829	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
10	9	biimpar 297	. 2 |- ( ( ( A e. V /\ B e. W ) /\ E. x ( A D x /\ x C B ) ) -> A ( C o. D ) B )
11	1, 2, 8, 10	syl21anc 1248	1 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   class class class wbr 4029   o. ccom 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-co 4668

This theorem is referenced by: (None)